August 06, 2018 Monday
Bedtime Story
What Abel said of Ruffini
Last night I had introduced you two
somewhat contradictory theorems of algebra:
The first was the Abel-Ruffini theorem
The second was what is nowadays known as
the fundamental theorem of algebra
Whereas the first one talks about unsolvability
(general and in radicals) of polynomials of degree five and higher the second
one emphatically states that every non-constant polynomial equation has at
least complex number as solution.
And who do you think provided this proof of
the so called fundamental theorem of algebra.
It was once again Gauss of course.
Thus every non-constant single variable
polynomial with complex coefficients has at least one complex root.
Hence it is important that one gets the
Abel-Ruffini theorem right.
The Abel-Ruffini theorem also does not
proclaim that no higher-degree polynomial equations can be solved by radicals,
for that is certainly possible.
The theorem merely proves that there is no
general solution in radicals that applies to all equations (in a generalized
form) of a given degree greater than four.
As we saw with the quadratic, cubic and
quartic equations, there was a general solution even though the solution to the
quartic equation is hugely cumbersome and unwieldy to be of use by most
mathematicians.
When Ruffini proved his theorem, as an
honor to the person who helped initiate his work, he sent his proof to Lagrange
in 1799 but for some reasons unknown both to him and me, Ruffini received no
reply from Lagrange.
Ruffini felt that his work was important
and so he also sent his proof to several other mathematicians one amongst whom
was the French mathematician Augustin-Louis Cauchy (and a prolific writer) who
wrote back to Ruffini:
“Your memoir on the general solution of
equations is a work which I have always believed should be kept in mind by
mathematicians and which, in my opinion, proves conclusively the algebraic
unsolvability of general equations higher than fourth degree.”
This as we know turned out to not to be
entirely correct as the proof turned out to be incomplete and hence could not
be considered a proof.
When Niels Abel went through the proof of
Ruffini he commented:
“The first and, if I am not mistaken, the
only one who, before me, has sought to prove the impossibility of the algebraic
solution of general equations is the mathematician Ruffini.
But his memoir is so complicated that it is
very difficult to determine the validity of his argument.
It seems to me that his argument is not
completely satisfying.”
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night Mon Ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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